Klann and Ramlau [16] hypothesized fractional Tikhonov regularization as an interpolation between generalized inverse and Tikhonov regularization. In fact, fractional schemes can be viewed as a generalization of the Tikhonov scheme. One of the motives of this work is the major pitfall of the a priori parameter choice rule, which primarily relies on source conditions that are often unknown. It necessitates the need for advocating a data-driven approach (a posteriori choice strategy). We briefly overview fractional scheme in learning theory and propose a modified Engl type [9] discrepancy principle, thus integrating supervised learning into the field of inverse problems. In due course of the investigation, we effectively explored the relation between learning from examples and the inverse problems. We demonstrate the regularization properties and establish the convergence rate of this scheme. Finally, the theoretical results are corroborated using two well known examples in learning theory.

Regularization is a method for providing a stable approximate solution to ill-posed operator equations, and it involves the regularization parameter which plays an important role in the convergence of the method. In this article, we propose a class of a posteriori parameter choice rules for filter-based regularization methods and establish the optimal rate of convergence O(?) from the proposed rules. We study these methods along with the proposed parameter choice rules in the context of pseudo-differential operator equations as well as the analytic continuation problem. The numerical implementation of the pseudo-differential operator equation and analytic continuation problem is discussed.

We explore the further study of filter based schemes along with parameter choice rules and then apply to pseudo-differential operator equations and the analytic continuation problem. As we know that the stability of the regularization schemes depends on the regularization parameter and here we derive an a posteriori parameter choice rule which is best among all other parameter choice rules; and this parameter choice rule minimizes an upper bound of the approximation error ?x?,??x†?. Numerical experiments are also provided to validate the proposed theory. © 2024 Elsevier Inc.

Recently, Reddy (2018) has studied a posteriori parameter choice rules for the weighted Tikhonov regularization scheme. In this article, we consider the finite dimensional weighted Tikhonov scheme and discuss the convergence analysis of the scheme. We introduce a class of parameter choice strategies to choose the regularization parameter in the finite dimensional weighted Tikhonov scheme; and derive the optimal rate of convergence O(??+1?+2) for the scheme based on the proposed strategies. Results of numerical experiments are documented to illustrate the performance of the scheme with an efficient finite dimensional approximation of an operator.

Many problems in science and engineering can be modeled as singularly perturbed partial differential equations. Solutions to such problems are not generally continuous with respect to the perturbation parameter(s) and hence developing stable and efficient numerical schemes to singularly perturbed problems are always more interesting and mathematically challenging to researchers. In this paper, we propose a generalized regularization scheme as an alternate numerical scheme for solving singularly perturbed parabolic PDEs with higher-order derivatives multiplied by a small parameter. Since the involved operators are unbounded, first, we propose a general theoretical framework for unbounded operators with an a posteriori parameter choice rule for selecting a regularization parameter, and then we apply it to singularly perturbed problems. We numerically implement the proposed scheme and compare it with other traditional schemes. The study illustrates that the proposed scheme can be considered as an alternate and competent method for solving singularly perturbed parabolic problems.

Singularly perturbed partial differential equations occur in many practical problems. These equations are characterized mathematically by the presence of a small parameter ? multiplying to one or more of the highest derivatives in a partial differential equation. Finding stable approximate solutions to such problems is mathematically challenging and interesting due to the fact that the solution is not stable as ? goes to 0. Many numerical schemes have been explored in the literature for finding approximate solutions to such PDEs. However, in this paper, our attempt is to propose a regularized iterative scheme as an alternative approach for finding stable approximate solutions to singularly perturbed elliptic and parabolic PDEs. We propose a general theoretical frame work for unbounded operators and then apply it for solving both singularly perturbed parabolic and elliptic PDEs. Theoretical results are illustrated through numerical examples and compared with other standard schemes. Numerical investigation assert that the proposed scheme is a very competitive and an alternate approach for solving the singularly perturbed problems.